Without solving, how do you determine the number of solutions for the equation #sectheta=(2sqrt3)/3# in #-180^circ<=theta<=180^circ#?

1 Answer
Feb 18, 2018

Answer:

Two solutions.
See explanation.

Explanation:

#sec theta = 1/(cos theta)=((2sqrt3)/3)#

So you're really looking at where
#cos(theta)=(3)/(2sqrt3)=(sqrt3)/2# (looks more familiar?)

#-180^@ = -pi# radian
#180^@ = pi# radian

Since the cosine function is "even", #cos (-x) = cos(x)# for all #x#.
So if there is a solution from 0 to #pi#, there is also a solution from 0 to #-pi#.

We can readily see that #0 < sqrt(3)/2 < 1#,
because #sqrt(3) < sqrt(4)#.
We know that the range of the cosine function is between -1 and 1, so this is good news, this means that there is a solution (if the value was larger than 1, we would have had zero solution).

Therefore,
#cos(theta)=sqrt(3)/2# has a solution that is between (0 and #180^@#), and another solution that is between (0 and #-180^@#).
Thus,
#sec theta= ((2sqrt3)/3)# has two solutions in the range #-180^@ < theta < 180^@#.
Q.E.D.